/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.pl /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.pl # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Left Termination of the query pattern goal(g,a,a) w.r.t. the given Prolog program could successfully be proven: (0) Prolog (1) PrologToPiTRSProof [SOUND, 14 ms] (2) PiTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) PiDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) AND (7) PiDP (8) UsableRulesProof [EQUIVALENT, 0 ms] (9) PiDP (10) PiDPToQDPProof [SOUND, 0 ms] (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) PiDP (15) UsableRulesProof [EQUIVALENT, 0 ms] (16) PiDP (17) PiDPToQDPProof [SOUND, 0 ms] (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) PiDP (22) UsableRulesProof [EQUIVALENT, 0 ms] (23) PiDP (24) PiDPToQDPProof [SOUND, 0 ms] (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) YES ---------------------------------------- (0) Obligation: Clauses: goal(A, B, C) :- ','(s2l(A, D), applast(D, B, C)). applast(L, X, Last) :- ','(append(L, .(X, []), LX), last(Last, LX)). last(X, .(X, [])). last(X, .(H, T)) :- last(X, T). append([], L, L). append(.(H, L1), L2, .(H, L3)) :- append(L1, L2, L3). s2l(s(X), .(Y, Xs)) :- s2l(X, Xs). s2l(0, []). Query: goal(g,a,a) ---------------------------------------- (1) PrologToPiTRSProof (SOUND) We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes: goal_in_3: (b,f,f) s2l_in_2: (b,f) applast_in_3: (b,f,f) append_in_3: (b,b,f) last_in_2: (f,b) Transforming Prolog into the following Term Rewriting System: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog ---------------------------------------- (2) Obligation: Pi-finite rewrite system: The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2l_in_ga(A, D)) GOAL_IN_GAA(A, B, C) -> S2L_IN_GA(A, D) S2L_IN_GA(s(X), .(Y, Xs)) -> U7_GA(X, Y, Xs, s2l_in_ga(X, Xs)) S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) U1_GAA(A, B, C, s2l_out_ga(A, D)) -> U2_GAA(A, B, C, applast_in_gaa(D, B, C)) U1_GAA(A, B, C, s2l_out_ga(A, D)) -> APPLAST_IN_GAA(D, B, C) APPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, append_in_gga(L, .(X, []), LX)) APPLAST_IN_GAA(L, X, Last) -> APPEND_IN_GGA(L, .(X, []), LX) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U6_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3)) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) U3_GAA(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_GAA(L, X, Last, last_in_ag(Last, LX)) U3_GAA(L, X, Last, append_out_gga(L, .(X, []), LX)) -> LAST_IN_AG(Last, LX) LAST_IN_AG(X, .(H, T)) -> U5_AG(X, H, T, last_in_ag(X, T)) LAST_IN_AG(X, .(H, T)) -> LAST_IN_AG(X, T) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa GOAL_IN_GAA(x1, x2, x3) = GOAL_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x4) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) U7_GA(x1, x2, x3, x4) = U7_GA(x4) U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) APPLAST_IN_GAA(x1, x2, x3) = APPLAST_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x5) U4_GAA(x1, x2, x3, x4) = U4_GAA(x4) LAST_IN_AG(x1, x2) = LAST_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (4) Obligation: Pi DP problem: The TRS P consists of the following rules: GOAL_IN_GAA(A, B, C) -> U1_GAA(A, B, C, s2l_in_ga(A, D)) GOAL_IN_GAA(A, B, C) -> S2L_IN_GA(A, D) S2L_IN_GA(s(X), .(Y, Xs)) -> U7_GA(X, Y, Xs, s2l_in_ga(X, Xs)) S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) U1_GAA(A, B, C, s2l_out_ga(A, D)) -> U2_GAA(A, B, C, applast_in_gaa(D, B, C)) U1_GAA(A, B, C, s2l_out_ga(A, D)) -> APPLAST_IN_GAA(D, B, C) APPLAST_IN_GAA(L, X, Last) -> U3_GAA(L, X, Last, append_in_gga(L, .(X, []), LX)) APPLAST_IN_GAA(L, X, Last) -> APPEND_IN_GGA(L, .(X, []), LX) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> U6_GGA(H, L1, L2, L3, append_in_gga(L1, L2, L3)) APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) U3_GAA(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_GAA(L, X, Last, last_in_ag(Last, LX)) U3_GAA(L, X, Last, append_out_gga(L, .(X, []), LX)) -> LAST_IN_AG(Last, LX) LAST_IN_AG(X, .(H, T)) -> U5_AG(X, H, T, last_in_ag(X, T)) LAST_IN_AG(X, .(H, T)) -> LAST_IN_AG(X, T) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa GOAL_IN_GAA(x1, x2, x3) = GOAL_IN_GAA(x1) U1_GAA(x1, x2, x3, x4) = U1_GAA(x4) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) U7_GA(x1, x2, x3, x4) = U7_GA(x4) U2_GAA(x1, x2, x3, x4) = U2_GAA(x4) APPLAST_IN_GAA(x1, x2, x3) = APPLAST_IN_GAA(x1) U3_GAA(x1, x2, x3, x4) = U3_GAA(x4) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) U6_GGA(x1, x2, x3, x4, x5) = U6_GGA(x5) U4_GAA(x1, x2, x3, x4) = U4_GAA(x4) LAST_IN_AG(x1, x2) = LAST_IN_AG(x2) U5_AG(x1, x2, x3, x4) = U5_AG(x4) We have to consider all (P,R,Pi)-chains ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes. ---------------------------------------- (6) Complex Obligation (AND) ---------------------------------------- (7) Obligation: Pi DP problem: The TRS P consists of the following rules: LAST_IN_AG(X, .(H, T)) -> LAST_IN_AG(X, T) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa LAST_IN_AG(x1, x2) = LAST_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (8) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (9) Obligation: Pi DP problem: The TRS P consists of the following rules: LAST_IN_AG(X, .(H, T)) -> LAST_IN_AG(X, T) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) LAST_IN_AG(x1, x2) = LAST_IN_AG(x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (10) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: LAST_IN_AG(.(T)) -> LAST_IN_AG(T) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (12) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LAST_IN_AG(.(T)) -> LAST_IN_AG(T) The graph contains the following edges 1 > 1 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (15) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (16) Obligation: Pi DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(H, L1), L2, .(H, L3)) -> APPEND_IN_GGA(L1, L2, L3) R is empty. The argument filtering Pi contains the following mapping: .(x1, x2) = .(x2) APPEND_IN_GGA(x1, x2, x3) = APPEND_IN_GGA(x1, x2) We have to consider all (P,R,Pi)-chains ---------------------------------------- (17) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: APPEND_IN_GGA(.(L1), L2) -> APPEND_IN_GGA(L1, L2) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (19) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *APPEND_IN_GGA(.(L1), L2) -> APPEND_IN_GGA(L1, L2) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Pi DP problem: The TRS P consists of the following rules: S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) The TRS R consists of the following rules: goal_in_gaa(A, B, C) -> U1_gaa(A, B, C, s2l_in_ga(A, D)) s2l_in_ga(s(X), .(Y, Xs)) -> U7_ga(X, Y, Xs, s2l_in_ga(X, Xs)) s2l_in_ga(0, []) -> s2l_out_ga(0, []) U7_ga(X, Y, Xs, s2l_out_ga(X, Xs)) -> s2l_out_ga(s(X), .(Y, Xs)) U1_gaa(A, B, C, s2l_out_ga(A, D)) -> U2_gaa(A, B, C, applast_in_gaa(D, B, C)) applast_in_gaa(L, X, Last) -> U3_gaa(L, X, Last, append_in_gga(L, .(X, []), LX)) append_in_gga([], L, L) -> append_out_gga([], L, L) append_in_gga(.(H, L1), L2, .(H, L3)) -> U6_gga(H, L1, L2, L3, append_in_gga(L1, L2, L3)) U6_gga(H, L1, L2, L3, append_out_gga(L1, L2, L3)) -> append_out_gga(.(H, L1), L2, .(H, L3)) U3_gaa(L, X, Last, append_out_gga(L, .(X, []), LX)) -> U4_gaa(L, X, Last, last_in_ag(Last, LX)) last_in_ag(X, .(X, [])) -> last_out_ag(X, .(X, [])) last_in_ag(X, .(H, T)) -> U5_ag(X, H, T, last_in_ag(X, T)) U5_ag(X, H, T, last_out_ag(X, T)) -> last_out_ag(X, .(H, T)) U4_gaa(L, X, Last, last_out_ag(Last, LX)) -> applast_out_gaa(L, X, Last) U2_gaa(A, B, C, applast_out_gaa(D, B, C)) -> goal_out_gaa(A, B, C) The argument filtering Pi contains the following mapping: goal_in_gaa(x1, x2, x3) = goal_in_gaa(x1) U1_gaa(x1, x2, x3, x4) = U1_gaa(x4) s2l_in_ga(x1, x2) = s2l_in_ga(x1) s(x1) = s(x1) U7_ga(x1, x2, x3, x4) = U7_ga(x4) 0 = 0 s2l_out_ga(x1, x2) = s2l_out_ga(x2) .(x1, x2) = .(x2) U2_gaa(x1, x2, x3, x4) = U2_gaa(x4) applast_in_gaa(x1, x2, x3) = applast_in_gaa(x1) U3_gaa(x1, x2, x3, x4) = U3_gaa(x4) append_in_gga(x1, x2, x3) = append_in_gga(x1, x2) [] = [] append_out_gga(x1, x2, x3) = append_out_gga(x3) U6_gga(x1, x2, x3, x4, x5) = U6_gga(x5) U4_gaa(x1, x2, x3, x4) = U4_gaa(x4) last_in_ag(x1, x2) = last_in_ag(x2) last_out_ag(x1, x2) = last_out_ag U5_ag(x1, x2, x3, x4) = U5_ag(x4) applast_out_gaa(x1, x2, x3) = applast_out_gaa goal_out_gaa(x1, x2, x3) = goal_out_gaa S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (22) UsableRulesProof (EQUIVALENT) For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R. ---------------------------------------- (23) Obligation: Pi DP problem: The TRS P consists of the following rules: S2L_IN_GA(s(X), .(Y, Xs)) -> S2L_IN_GA(X, Xs) R is empty. The argument filtering Pi contains the following mapping: s(x1) = s(x1) .(x1, x2) = .(x2) S2L_IN_GA(x1, x2) = S2L_IN_GA(x1) We have to consider all (P,R,Pi)-chains ---------------------------------------- (24) PiDPToQDPProof (SOUND) Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi. ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: S2L_IN_GA(s(X)) -> S2L_IN_GA(X) R is empty. Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (26) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *S2L_IN_GA(s(X)) -> S2L_IN_GA(X) The graph contains the following edges 1 > 1 ---------------------------------------- (27) YES